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This is an exercise from A Modern Formal Logic Primer (Teller):

4-1. Give examples, using sentences in English, of arguments of each of the following kind. Use examples in which it is easy to tell whether the premises and the conclusion are in fact (in real life) true or false.

e) An invalid argument with one or more false premises

Validity of an argument has been defined as follows:

The argument "$X$. Therefore $Y$" is valid just in case the sentence $\neg (X \land \neg Y)$ is a logical truth.

Now my confusion is twofold:

  1. An invalid argument with one or more false premises appears to require that at least one premise and the conclusion are contradictions. Otherwise there are cases in which all premises are true. But for the argument to be invalid, there must be at least one case in which the conjunction of the premises is true while the conclusion is false. How is this possible?

  2. I am not sure how to interpret "logical truth" in terms of English sentences with invariable truth values. Here is the solution given in the book:

Anyone who loves logic is bald.
Robert Redford is bald.

Therefore Redford loves logic.

As far as I can tell, both of the premises are false (is it correct to interpret invariably false natural language sentences as contradictions?). Which makes $\neg (X \land \neg Y)$ true. Now since the truth value of the premises is invariable, there is no case in which the conjunction of the premises is true while the conclusion is false. So how is this an invalid argument?

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An invalid argument does not require any contradictions at all. Example: 1. Unicorns exist. 2. It rains everywhere on the Earth at all times. Therefore, 3. I am flying to San Francisco.

Your second example commits the fallacy of the undistributed middle. You haven't talked about all bald people, so it can't be the middle term in an Aristotelian syllogism.

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  • $\begingroup$ I don't see how your argument is invalid under the definition given in the book: An argument is invalid given an assignment of truth values to sentence letters which makes the conjunction of premises true and the conclusion false. $\endgroup$ – user245312 Jan 15 at 17:48
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    $\begingroup$ The book is asking you to provide an invalid argument with false premises. I have provided you an invalid argument with false premises. It's invalid because if the premises WERE true, they would not force the conclusion to be true. $\endgroup$ – Adrian Keister Jan 15 at 18:32
  • $\begingroup$ I see, if the premises are independent from the conclusion then $\neg (X \land \neg Y)$ cannot be a logical truth. $\endgroup$ – user245312 Jan 15 at 19:39

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